Sets

What is a set

sets are the fundamental discrete structure on which all the discrete structures are built. sets are used to group objects together.

formally speaking, An unordered, well defined, collection of distinct objects is called a set. the object of set is called as element or member of set

example:

$A = \{0, 2, 4, 8\}$
$B = \{1, 3, 5\}$
$C = \{x \mid x \in \mathbb{N}\}$

Forms of Representing a Set

Set is denoted by capital letter (S). The objects of set is denoted with small letters Generally the sets are represented in two ways

1. Tabular Form (Roster Form)

   • In the tabular (or roster) form, all the elements of a set are listed inside curly braces {} and separated by commas.
   • This method is suitable when the set contains a small number of elements.
   • Example: $A = \{0, 2, 4, 8\}$

2. Set-Builder Form

   • In the set-builder form, a set is described by stating the common property or rule that all its elements satisfy.
   • This method is useful for representing large or infinite sets.
   • Example: $C = \{x \mid x \in \mathbb{N}\}$

basic terms of sets

cardinality

• the number of elements in the set is called as the cardinality of the set
• it is represented by $|A|$
• ex: $A = \{1, 2, 3, 4\}$ the cardinality of set A is 4.

subset

• let there are 2 sets $A, B$. set $A$ is said to be subset of $B$ if and only if all the elements of $A$ is present in the set $B$
• it is represented as $A \subseteq B$

proper subset

• if $A$ is subset of $B$ and $A \neq B$, then $A$ is the proper subset of $B$.
• proper subset is denoted by $A \subset B$.